The consequences of this addition are twofold. First, the global scale \(\tau \)  shrinks all the weights towards zero, encouraging the model to learn sparse representations of the underlying factors. This is especially useful when learning sparse unannotated factors. Second, the local scales act as a regulator on each individual weight. A small local scale \(\lambda _{g, k}\)  confines \(w_{g, k}\)  to values very close to zero. On the other hand, a large local scale allows its corresponding parameter to escape the global regularization penalty. As an enhancement to the original horseshoe prior[1]}, the regularized version provides means to control the slab width of each individual scale[2]}, allowing partial knowledge to guide the model, e.g. annotated gene sets.
\(w_{g, k} | \tau , \lambda _{g, k}, c_{g, k} \sim \mathcal {N}(0, \tau ^2 \cdot \tilde{\lambda }_{g, k}^2), \quad \tilde{\lambda }_{g, k} = \frac{c_{g, k}^2\lambda _{g, k}^2}{c_{g, k}^2 + \tau ^2\lambda _{g, k}^2}\) 
